The precise one-shot characterisation of operational tasks in classical and quantum information theory relies on different forms of smooth entropic quantities. A particularly important connection is between the hypothesis testing relative entropy and the smoothed max-relative entropy, which together govern many operational settings. We first strengthen this connection into a type of equivalence: we show that the hypothesis testing relative entropy is equivalent to a variant of the smooth max-relative entropy based on the information spectrum divergence, which can be alternatively understood as a measured smooth maxrelative entropy. Furthermore, we improve a fundamental lemma due to Datta and Renner that connects the different variants of the smoothed max-relative entropy, introducing a modified proof technique based on matrix geometric means. We use the unveiled connections and tools to strictly improve on previously known one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, sharpening also bounds that connect the max-relative entropy with Rényi divergences.

We study a variant of quantum hypothesis testing wherein an additional ‘inconclusive’ measurement outcome is added, allowing one to abstain from attempting to discriminate the hypotheses. The error probabilities are then conditioned on a successful attempt, with inconclusive trials disregarded. We completely characterise this task in both the single-shot and asymptotic regimes, providing exact formulas for the optimal error probabilities. In particular, we prove that the asymptotic error exponent of discriminating any two quantum states$\rho $and$\sigma $is given by the Hilbert projective metric$D_{\max }(\rho \|\sigma ) + D_{\max }(\sigma \| \rho )$in asymmetric hypothesis testing, and by the Thompson metric$\max \! \big \{ D_{\max }(\rho \|\sigma ),\, D_{\max }(\sigma \| \rho ) \big \}$in symmetric hypothesis testing. This endows these two quantities with fundamental operational interpretations in quantum state discrimination. Our findings extend to composite hypothesis testing, where we show that the asymmetric error exponent with respect to any convex set of density matrices is given by a regularisation of the Hilbert projective metric. We apply our results also to quantum channels, showing that no advantage is gained by employing adaptive or even more general discrimination schemes over parallel ones, in both the asymmetric and symmetric settings. Our state discrimination results make use of no properties specific to quantum mechanics and are also valid in general probabilistic theories.

The distillable randomness of a bipartite quantum state is an information-theoretic quantity equal to the largest net rate at which shared randomness can be distilled from the state by means of local operations and classical communication. This quantity has been widely used as a measure of classical correlations, and one version of it is equal to the regularized Holevo information of the ensemble that results from measuring one share of the state. However, due to the regularization, the distillable randomness is difficult to compute in general. To address this problem, we define measures of classical correlations and prove a number of their properties, most importantly that they serve as upper bounds on the distillable randomness of an arbitrary bipartite state. We then further bound these measures from above by some that are efficiently computable by means of semi-definite programming, we evaluate one of them for the example of an isotropic state, and we remark on the relation to quantities previously proposed in the literature.Full version at https://markwilde.com/RD-bnds.pdf